The Geometry of Discrete Groups
The Geometry of Discrete Groups
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DOI：
10.1007/978-1-4612-1146-4
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发表时间：
1995-09
期刊：
影响因子：
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通讯作者：
A. Beardon
中科院分区：
文献类型：
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作者：
A. Beardon
这篇文章的目的是作为一个介绍莫比乌斯变换的离散群的行动几何。一百多年来，人们对这一主题的研究重点不断变化，最近的发展与3流形理论有关：例如，参见庞加莱[77]和瑟斯顿[101]的论文。大约在1940年，芬切尔-尼尔森的手稿出现了。遗憾的是，这份手稿从未出版过，而这篇较为朴素的文本试图展示至少一些在手稿中发现的美丽的地理几何思想，以及一些更近期的材料。本书的写作信念是，几何解释对于充分理解材料至关重要，而且无论矩阵证明看起来多么简单，几何证明几乎肯定更有益。此外，只要有可能，结果应该以一种在共轭下不变的形式来表述，从而使结果的内在本质更加明显。尽管事实上主题是与双曲几何的等距组有关，但许多出版物依赖于欧几里得估计和几何。然而，最近的发展再次强调了对双曲几何的需要，我已经包含了一个关于解析（不是公理化）双曲几何的综合章节。我们希望这一章能成为平面双曲几何公式的“字典”，从而使人们对其本身感兴趣并有所应用。
This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a" dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.